lecture07

lecture07 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 1 SensitivityAnalyses YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 2 ParametricLinearProgrammingProblem The objectivecoefficient vectorbecomes c + λ g and/orthe right-hand-side vectorbecomes b + λ d ,wheretheparameter λ belongstoan interval . DenotethisproblembyLP( λ ): LP ( λ ) minimize ( c + λ g ) T x subjectto A x = b + λ d , x ≥ . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 3 GeometricalObservations 1.Weknowthatforthefunction c T x ,thevector c denotesthe directionof steepestascent .Thus, parameterizing thecostfunctionaccordingtotherule c + λ g changesthe gradientorslope oftheobjectivehyperplane.This raisesthequestionofwhetherornotthecurrent basis B remainsoptimal . 2.If b isreplacedby b + λ d ,as λ varyingthepoint b + λ d movesawayfrom b inthedirection d (dependingonthesignof λ ).Thisraisesthequestionof whetherornotthenewRHS b + λ d hasthecurrent basis B remains feasible ,thatis, A B x B = b + λ d implies x B ≥ . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 4 RecallWhenaBSFisoptimal GivenaBFSintheLPstandardform A B x B = b , x B ≥ , x N = , anditscompanionshadowprice(dualsolution)andreducedcostvectors: A T B y = c B , ( and r = c- A T y ) . Ifthereducedcostvector r ≥ ,thentheBFSwithbasicvariableset B ,thenthe BFSis optimal .(–Whataboutthe maximization ?) Atoptimalitywealwayshave c T x = b T y . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 5 GettingStarted Letusconsider λ varyingaround . Akeyquestionintheseparametricproblemsis:howmuchcantheparameter λ bechangedbeforethecurrent optimalbasis B ofLP( )islost? Theorem1 TheoptmalbasisofLP( )remainsoptimalforLP( λ )ifandonlyif A- 1 B ( b + λ d ) ≥ and ( c + λ g )- A T ( A TB )- 1 ( c + λ g ) B ≥ . Thiswillestablishanintervalon...
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lecture07 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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